PDE Solving

PDE: Partial Differential Equation

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Overview

PDE solving seeks approximate solutions to equations describing processes such as heat flow, fluids, waves, and diffusion, where exact solutions are often unavailable. Classical approaches include finite difference, finite element, finite volume, spectral, and domain-decomposition methods; learning-based approaches include physics-informed neural networks, neural operators, and randomized or basis-based solvers. Current work also combines these methods, using learned models for acceleration, adaptive refinement, or surrogate prediction while retaining numerical constraints such as conservation and boundary conditions. A central challenge is balancing speed and flexibility with accuracy, stability, reliable generalization, and guarantees that residual errors reflect true solution errors.

Latest papers

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255 papers30 papers in the last four weeks, up 20% on the four weeks before. 0.2% of all new papers.

  1. A Coupled Fourth Order Telegraph Diffusion Framework Using Grayscale Indicators for Image Despeckling

    Apr 26, 2026Manish Kumar, Rajendra K. RayPDE SolvingSynthetic Aperture Radar

    School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi (H.P), 175075, India

  2. Single Image Defogging Using a Fourth-Order Telegraph PDE Guided by Physical Haze Modeling

    Apr 26, 2026Manish Kumar, Rajendra K. RayPDE SolvingPhysics-Guided Image Restoration

    School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi (H.P), 175075, India

  3. When PINNs Go Wrong: Pseudo-Time Stepping Against Spurious Solutions

    Apr 26, 2026Sifan Wang, Shawn Koohy, Yiping Lu +1PDE SolvingPhysics-Informed ML

    Institute for Foundations of Data Science, Yale University · Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania · Industrial Engineering & Management Science, Northwestern University

  4. Transferable Physics-Informed Representations via Closed-Form Head Adaptation

    Apr 23, 2026Jian Cheng Wong, Isaac Yin Chung Lai, Pao-Hsiung Chiu +3Representation LearningPDE Solving

    Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore · National University of Singapore (NUS), Singapore · Indian Institute of Technology Goa (IIT Goa), India +1

  5. A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization

    Apr 23, 2026Mohammad Mahdi Abedi, David Pardo, Tariq AlkhalifahNeural PDE SolversPDE Solving

    University of the Basque Country (UPV/EHU), Department of Mathematics, Spain · Basque Center for Applied Mathematics (BCAM), Bilbao, Spain · Ikerbasque, Basque Foundation for Science, Bilbao, Spain +1

  6. CO2_2 sequestration hybrid solver using isogeometric alternating-directions and collocation-based robust variational physics informed neural networks (IGA-ADS-CRVPINN)

    Apr 22, 2026Askold Vilkha, Tomasz Służalec, Marcin Łoś +1Neural PDE SolversPorous Media Flow

    AGH University of Krakow, Faculty of Computer Science, Al. Mickiewicza 2026 30, Kraków, 30-059, Poland

  7. Virtual boundary integral neural network for three-dimensional exterior acoustic problems

    Apr 19, 2026Jiahao Li, Qiang Xi, Ilia Marchevskiy +1Neural PDE SolversPDE Solving

    College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China · Department of Applied Mathematics, Bauman Moscow State Technical University, Moscow 105005, Russia

  8. Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches

    Apr 18, 2026Yixuan WangPDE SolvingKolmogorov-Arnold Networks

    CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California · [math.NA]

  9. A Randomized PDE Energy driven Iterative Framework for Efficient and Stable PDE Solutions

    Apr 17, 2026Yi Bing, Zheng Ran, Fu Jinyang +2PDE Solving

    School of Traffic and Transportation Engineering, Central South University, Changsha, China · School of Civil Engineering, Central South University, Changsha, China · National Engineering Research Center of High-speed Railway Construction Technology, Changsha, China +2

  10. A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws

    Apr 17, 2026Jiamin Jiang, Shanglin Lv, Jingrun ChenGraph Neural NetworksNeural PDE Solvers

    School of Artificial Intelligence and Data Science and Suzhou Institute for Advanced Research, University of Science and Technology of China, and Suzhou Big Data & AI Research and Engineering Center, China · School of Mathematical Sciences, University of Science and Technology of China, China · School of Mathematical Sciences and Suzhou Institute for Advanced Research, University of Science and Technology of China, and Suzhou Big Data & AI Research and Engineering Center, China

  11. Python library supporting Discrete Variational Formulations and training solutions with Collocation-based Robust Variational Physics Informed Neural Networks (DVF-CRVPINN)

    Apr 16, 2026Tomasz Służalec, Marcin Łoś, Askold Vilkha +1PDE SolvingPhysics-Informed ML

    AGH University of Krakow, Poland

  12. Physics-Informed Neural Networks for Solving Derivative-Constrained PDEs

    Apr 15, 2026Kentaro Hoshisashi, Carolyn E Phelan, Paolo BaruccaNeural PDE SolversPDE Solving

    Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK

  13. Comparative Study of Bending Analysis using Physics-Informed Neural Networks and Numerical Dynamic Deflection in Perforated nanobeam

    Apr 15, 2026Ramanath Garai, Iswari Sahu, S. ChakravertyStructural MechanicsPhysics-Informed ML

    Department of Mathematics, National Institute of Technology Rourkela, Rourkela-769008, India.

  14. SCNO: Spiking Compositional Neural Operator -- Towards a Neuromorphic Foundation Model for Nuclear PDE Solving

    Apr 13, 2026Samrendra Roy, Souvik Chakraborty, Rizwan-uddin +1PDE SolvingPDE Surrogate Modeling

    University of Illinois Urbana-Champaign Department of Nuclear, Plasma & Radiological Engineering Urbana, IL, USA · Indian Institute of Technology Delhi Department of Applied Mechanics New Delhi, India · National Center for Supercomputing Applications Urbana, IL, USA

  15. Generative Path-Finding Method for Wasserstein Gradient Flow

    Apr 13, 2026Chengyu Liu, Xiang ZhouFokker–Planck EquationWasserstein Gradient Flows

    Department of Data Science, City University of Hong Kong, Kowloon, Hong Kong SAR · Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR

  16. Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains

    Apr 10, 2026Zhangyong LiangNeural PDE SolversPDE Solving

    National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China.

  17. Meta-Learned Basis Adaptation for Parametric Linear PDEs

    Apr 10, 2026Vikas Dwivedi, Monica Sigovan, Bruno SixouPDE SolvingPDE Surrogate Modeling

    CREATIS Biomedical Imaging Laboratory INSA, CNRS UMR 5220, Inserm, Université Lyon 1 Lyon 69621, France

  18. Physics-Informed Functional Link Constrained Framework with Domain Mapping for Solving Bending Analysis of an Exponentially Loaded Perforated Beam

    Apr 8, 2026Iswari Sahu, Ramanath Garai, S. ChakravertyPhysics-Informed MLStructural Mechanics

    Department of Mathematics, National Institute of Technology Rourkela, India

  19. Modelling Gas-Phase Reaction Kinetics with Guided Particle Diffusion Sampling

    Apr 8, 2026Andrew Millard, Zheng Zhao, Henrik PedersenPDE SolvingDiffusion Posterior Sampling

    Division of Statistics and Machine Learning (STIMA), Linköping University, Sweden · STIMA, Linköping University, Sweden · Division of Chemistry, Linköping University, Sweden

  20. A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributions

    Apr 7, 2026Xiaolong Wang, Jing Feng, Qi Liu +3PDE SolvingNeural PDE Solvers

    School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, China · School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China · MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, Xi’an, 710072, China +2

  21. Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies

    Apr 7, 2026Prashant Kumar, Rajesh RanjanDomain Decomposition MethodsNeural PDE Solvers

    Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016

  22. A Numerical Method for Coupling Parameterized Physics-Informed Neural Networks and FDM for Advanced Thermal-Hydraulic System Simulation

    Apr 3, 2026Jeesuk Shin, Donggyun Seo, Sihyeong Yu +1Physics-Based SimulationPDE Surrogate Modeling

    Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang-si, Republic of Korea · Mueunjae School of Undergraduate Studies, Pohang University of Science and Technology, Pohang-si, Republic of Korea · Department of Quantum System Engineering, Jeonbuk National University, Jeonju-si, Republic of Korea

  23. WGFINNs: Weak formulation-based GENERIC formalism informed neural networks

    Apr 3, 2026Jun Sur Richard Park, Auroni Huque Hashim, Siu Wun Cheung +2Physics-Informed MLAI-Assisted Scientific Research

    Division of Big Data Science, Korea University, Sejong, 30019, Republic of Korea · Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA · Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA.

  24. General Explicit Network (GEN): A novel deep learning architecture for solving partial differential equations

    Apr 2, 2026Genwei Ma, Ting Luo, Ping Yang +1Neural PDE SolversPDE Solving

    National Center for Applied Mathematics Beijing, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China · The Academy for Multidisciplinary Studies, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China · Academy of Information Network Security, People’s Public Security University of China, No.1 Muxidi Nanli, Beijing, 100038, China +2

  25. Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing

    Apr 2, 2026Yilong Dai, Shengyu Chen, Xiaowei Jia +1Neural PDE SolversPDE Solving

    The University of Alabama, Tuscaloosa, Alabama, USA · University of Pittsburgh, Pittsburgh, Pennsylvania, USA

  26. Macroscopic transport patterns of UAV traffic in 3D anisotropic wind fields: A constraint-preserving hybrid PINN-FVM approach

    Apr 1, 2026Hanbing Liang, Fujun LiuPDE SolvingPhysics-Informed ML

  27. Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes

    Mar 30, 2026Sean Disarò, Ruma Rani Maity, Aras BachoPDE SolvingPhysics-Informed ML

    Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India · Department of Computing and Mathematical Sciences, California Institute of Technology, CA, USA

  28. Quantum Neural Physics: Solving Partial Differential Equations on Quantum Simulators using Quantum Convolutional Neural Networks

    Mar 25, 2026Jucai Zhai, Muhammad Abdullah, Boyang Chen +6Neural PDE SolversPDE Solving

    Resource Geophysics Academy, Imperial College London, London, SW7 2AZ UK · Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London, SW7 2AZ UK · Quantum AI Leadership (QUAIL) Ltd, London, WC2H 9JQ UK +2

  29. Residual Attention Physics-Informed Neural Networks for Robust Multiphysics Simulation of Steady-State Electrothermal Energy Systems

    Mar 24, 2026Yuqing Zhou, Ze Tao, Fujun LiuPhysics-Informed MLPhysics-Based Simulation

    Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology, Changchun, 130022, P.R. China

  30. A Novel Method for Enforcing Exactly Dirichlet, Neumann and Robin Conditions on Curved Domain Boundaries for Physics Informed Machine Learning

    Mar 23, 2026Suchuan Dong, Yuchuan ZhangPDE SolvingPhysics-Informed ML

    Center for Computational and Applied Mathematics Department of Mathematics Purdue University, USA